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From Non-Model-Based to Model-Based Control ofPKMs: A Comparative Study

Hussein Saied, Ahmed Chemori, Maher El Rafei, Clovis Francis, FrançoisPierrot

To cite this version:Hussein Saied, Ahmed Chemori, Maher El Rafei, Clovis Francis, François Pierrot. From Non-Model-Based to Model-Based Control of PKMs: A Comparative Study. ICAMMRMS: International Congressfor the Advancement of Mechanism, Machine, Robotics and Mechatronics Sciences, Oct 2017, Bey-routh, Lebanon. �hal-01631006�

From Non-Model-Based to Model-Based Control of PKMs:

A Comparative Study

H. Saied1 A. Chemori2 M. El Rafei3 C. Francis4 F. Pierrot5

UM-LIRMM / LU-CRSI UM-LIRMM LU-CRSI LU-CRSI UM-LIRMM

Montpellier, France / Beirut, Lebanon Montpellier, France Beirut, Lebanon Beirut, Lebanon Montpellier, France

Abstract: This paper deals with control of parallel robots,

where different controllers are proposed and compared. It

demonstrates the strength of model-based controllers over

the non-model-based ones when dealing with parallel

kinematic manipulators known with their high nonlinearity,

time-varying parameters and uncertainties. More precisely,

adaptive model-based algorithms are the preferred control

solutions for such kind of manipulators, thanks to their

adjustable-parameters feature which is more adequate to

the varying and non-accurate nature of parallel kinematic

manipulators. These facts are fulfilled here by numerical

simulations and real-time experiments on a four-degree-of-

freedom parallel robot named VELOCE. Keywords: PKM, dynamic model, model-based, non-model-based,

adaptive control, feedforward, PID, Nonlinear PD, Augmented PD.

I. Introduction

Parallel kinematic manipulators (PKMs) are defined in

[1] as follows: “A generalized parallel manipulator is a

closed-loop kinematic chain mechanism whose end-

effector is linked to the base by several independent

kinematic chains”.

PKMs were extensively used in robotized industries in

the last few decades since it surpasses their counterpart’s

serial structures, particularly, in terms of high rigidity,

better tracking performance, good precision, high payload-

to-weight ratio and great dynamic [2], [3]. A very wide

range of applications take benefit of PKMs. Stewart

proposed in 1965 a platform that is used as a flight

simulator [4]. Delta robot prototype of 3 DOFs proposed

initially in 1985 [5] is the leader in pick-and-place

operations [1], used in packaging industry, laser cutting [6],

medical applications [7] and haptic devices in which they

allow the human-computer interaction [8]. Another parallel

structures are used in machining tasks [9].

However, some problems associated with such kind of

structures still open and are not solved satisfactory. The

drawbacks of PKMs are listed as limited range of motion

especially the rotational motion [10], small work space,

low dexterity, complex forward kinematic solutions [11].

Singularities’ behavior is more complicated than of serial

[12], it can occur both inside and on the border of the work

space [11].

In the literature, a wide range of control schemes have

been proposed aiming to drive PKMs in accurate mode and

high precision. The proposed control strategies can be

UM-LIRMM: University of Montpellier - Laboratoire

d’Informatique, de Robotique, et de Microélectronique de Montpellier. LU- CRSI: Lebanese University – Scientific Research Center in

Engineering.

{1hussein.saied, 2ahmed.chemori, 5francois.pierrot}@lirmm.fr {3maher.elrafei, 4cfrancis}@ul.edu.lb

classified in two classes, Model-Based and Non-Model-

Based. The non-model-based strategies do not need a priori

knowledge about the dynamics of the manipulator except

the states (position and velocity). The Proportional-

Integral-Derivative (PID) controller [13] is the most used

in industrial applications mainly due to its simplicity and

easy implementation as well as its acceptable control

performance. However, PKMs known by their nonlinear

dynamics, and highly increasing non-linearity at high

speeds which may even lead to instability. The need for

nonlinear controllers arises, knowing that PID lacks to

robustness. Nonlinear PD (NPD) controller [14] is more

adequate to the nature of PKMs, which can insure stability

and disturbances rejection and performs with better

robustness towards error variation. Successful application

of non-model-based fuzzy controller applied on Stewart

platform in [15], shows that this controller can drive the

six-degree motion platform accurately, smoothly and in a

stable way. On the other hand, researchers developed

several model-based controllers depending on the fact that

the closed-loop algorithms, rich enough with knowledge

about the system dynamics, can compensate their

nonlinearities. PD with gravity compensation or with

desired gravity compensation were applied intending to

achieve better performance than simple PD since it

surpasses the effect of gravity [16], [17]. Computed torque

(CT) control exploits the full knowledge about the

nonlinear system dynamics, leading to a linear closed-loop

system in terms of tracking error [18]. Also the Augmented

PD (APD) is a model-based strategy, where the dynamic

part of the controller is computed from both the desired and

measured states improving the global performance of the

control mission [19]. Nevertheless, PKMs are featured

with time-varying parameters (e.g. payload mass),

uncertainties and difficulty to get accurate model values,

then the design of adaptive controllers is very significant.

Adaptive model-based controllers recompense the possible

variation of parameters and react against the disturbances

by dynamical calibration in an online algorithm, such as

the adaptive feedforward PD controller (AFFPD) [20].

The control performances of some classical non-model-

based controllers, as PD, PID, NPD, and model-based

controllers, as APD, AFFPD, are studied and compared in

this paper. The main objective is to show that a controller

fed with a good dynamic knowledge about the robot will

be very powerful and more precise. Moreover, the time-

varying PKM environment requires adaptive dynamic

knowledge to manage robustness and accuracy, as it was

proved with real-time experimental tests.

The paper organization is as follow: Section II describes

the structure of VELOCE parallel robot, as well as its

kinematic and dynamic modeling. Section III is dedicated

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to the synthesis of the proposed control solutions.

Simulation and experimental results are presented and

discussed in section IV. Section V concludes the paper and

states the future work.

II. Description and Modeling of VELOCE PKM

In this section, a full description of the whole mechanical

structure of VELOCE PKM is presented, then a brief

explanation of its kinematic and dynamic models is

introduced.

A. Structure of VELOCE PKM

VELOCE robot (see Fig. 1) is a 4-DOF parallel

manipulator designed and fabricated in LIRMM

(Laboratoire d’Informatique, de Robotique et de

Microélectronique de Montpellier). It is mainly designed

for pick-and-place applications. It consists of four

kinematic chains and four degrees-of-freedom, three

independent translational degrees in the three dimensions

and one rotational degree around the vertical axis. It is note

that VELOCE is a fully parallel manipulator [1]. Each

kinematic chain is composed, in a serial manner, of an

actuator, a rear arm fixed to the actuator’s rotor, a forearm

including two links forming a parallelogram and connected

through ball joints to the rear arm and to the traveling plate

(see Fig. 2). The traveling plate is made of two essential

parts, upper and lower. Both parts are mounted on a single

screw, and the movement of one part with respect to the

other generates the rotational action.

B. Kinematic modeling of VELOCE PKM

The Cartesian coordinates of the traveling plate can be

presented with respect to the fixed-base frame in four-

dimensional space vector 𝒙 = [𝑥, 𝑦, 𝑧, 𝛼]𝑇 such that 𝑥, 𝑦, 𝑧

are the translational coordinates and 𝛼 is the rotational

angle around z-axis. The orientation and position of the

traveling plate are specified by the angular positions of the

four actuators, since VELOCE is a fully PKM, represented

in another four-dimensional space vector 𝒒 = [𝑞1, 𝑞2, 𝑞3, 𝑞4]𝑇. The relation between 𝒒 and 𝒙 is obtained

by a geometrical study for the constraints of the closed-

loop formed of kinematic chains and traveling plate. The

study leads to the following kinematic models, “Forward

kinematic ( 𝐹𝑘 )” and “Inverse kinematic ( 𝐼𝑘 )”

respectively: 𝒙 = 𝐹𝑘 (𝒒); 𝒒 = 𝐼𝑘 (𝒙) . Applying the

equiprojectivity principle explained in [3], the inverse

Fig. 1. VELOCE PKM. (a): CAD view, (b): The manufactured robot.

Jacobian matrix can be computed and thus a relation

between the joints’ velocities and Cartesian velocity of

traveling plate is formulated as follows:

�̇� = 𝑱𝒎�̇� (1)

where 𝑱𝒎 𝜖 ℝ4×4 is the inverse Jacobian Matrix. Note that

𝑱𝒎 is square and invertible for the fully PKMs (as

VELOCE), and if the chosen trajectory is away from

singularities. By differentiating equation (1) with respect to

time, we obtain the relation of accelerations between

Cartesian space and joint space as follows:

�̈� = 𝑱𝒎�̈� + �̇�𝒎�̇� (2)

C. Dynamic modeling of VELOCE PKM

According to [21], the dynamic model can be obtained by

analyzing the dynamics in the joint space and in the

traveling-plate space separately, then summing up the two

equations of motion. Nonetheless, some assumptions are

taken to simplify the complexity of the rigid body of such

robots. Standing on the light weight of the forearm, its

rotational inertia is neglected and its mass is split-up into

two parts, one part conjoined to the rear arm and one part

to the traveling plate mass. Also the dry and viscous friction

in the passive and active joints are ignored, and the effect

of gravity can be omitted at high speeds. Regarding the

traveling plate, there are three kind of forces acting on it:

the gravity forces, the inertial forces and the forces of the

load. These forces are transformed into contributions in

actuators’ torques using the Jacobian matrices. From the

joints side, the gravity of the rear arms with the half-masses

of forearms and the arms inertia are also expressed in the

actuators’ torques. Then, the total actuators’ torques vector

is computed by summing up the contributions of all forces.

One can formulate the total inverse dynamic equation to be

in the standard joint space form, so we get:

𝑴(𝒒)�̈� + 𝑪(𝒒, �̇�)�̇� + 𝑮(𝒒) + 𝜞𝑭𝒍𝒐𝒂𝒅= 𝜞 (3)

with 𝑴(𝒒) 𝜖 ℝ4×4 being the inertia matrix,

𝑪(𝒒, �̇�) 𝜖 ℝ4×4 is the Coriolis and centrifugal forces matrix,

𝑮(𝒒) 𝜖 ℝ4 be the gravitational forces vector and

𝜞𝑭𝒍𝒐𝒂𝒅 𝜖 ℝ4 be the payload forces vector. A fundamental

property of PKMs is very essential for model-based

adaptive controllers consists of linearity of the dynamics

with respect to the parameters, such as inertia and masses

[22]. So the reformulation of the dynamics in the linear

form is expressed as following:

𝒀(𝒒, �̇�, �̈�)𝜱 = 𝜞 (4)

where 𝒀(. ) 𝜖 ℝ4×𝑛 is the regression matrix which is

nonlinear function in terms of 𝒒, �̇� and �̈�, and 𝜱 𝜖 ℝ𝑛 is

the robot parameters vector to be estimated.

III. Proposed Control Solutions

In this section, a design for the proposed control

solutions is clarified. The controllers designed are non-

model-based controllers: PD, PID and NPD, and model-

based controllers: APD and AFFC. Note that the

available measurements are directly the joint angles of

the actuators, so all controllers are developed in joint

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The 1st International Congress for the Advancement of Mechanism, Machine, Robotics and Mechatronics Sciences, Beirut, Lebanon, 17-19 October, 2017

Fig. 2. An ith kinematic chain of VELOCE PKM

space.

A. Proportional-Derivative Controller

PD control scheme is composed of two parts, proportional

and derivative parts. The general expression of the control

input is:

𝜞 = 𝑲𝒑𝒆 + 𝑲𝒅�̇� (5)

where 𝒆 = 𝒒𝒅 − 𝒒 is the joint position error between the

desired angles and the actual measured ones.

𝑲𝒑, 𝑲𝒅 𝜖 ℝ4×4 are diagonal positive definite matrices

which means that no coupling between the joints is

considered, and the controller is called a linear single-axis

controller [2], knowing that the same gain is used for all

joints. The PD control law is asymptotically stable as was

addressed in [22]. It is the simplest control law but it has

several drawbacks briefed as weak disturbance rejection,

no compensation for the nonlinearity and variation nature,

and even may leads to instability at high accelerations.

B. Proportional-Integral-Derivative Controller

It is the same demonstration of the aforementioned PD

controller just adding the integral term which is the

multiplication of the integral of position error with a

positive constant feedback. The control law equation is

then:

𝜞 = 𝑲𝒑𝒆 + 𝑲𝒊 ∫ 𝒆 𝒅𝒕 + 𝑲𝒅�̇� (6)

where 𝑲𝒊 𝜖 ℝ4×4 is a diagonal positive definite matrix.

Same specifications and draw backs of the PD control, but

better global performance related to the tracking error

thanks to the contribution of integral term in eliminating

the residual errors in the steady state response produced by

the proportional term.

C. Nonlinear Proportional-Derivative Controller

This controller have the same structure of classical PD

controller with the time-varying feedback gains instead of

being constant. The feedback gains are nonlinear functions

in terms of the system states, inputs, and other variables.

As For the classical PD, the control law equation can be

written as following with the nonlinear gains functions [3]:

𝜞 = 𝑲𝒑 𝒇(𝒆, 𝛼1, 𝛿1) 𝒆 + 𝑲𝒅 𝒇(�̇�, 𝛼2, 𝛿2) �̇� (7)

Where 𝒇(𝒙, 𝛼, 𝛿) = {|𝒙|𝛼−1, |𝒙| > 𝛿

𝛿𝛼−1, |𝒙| ≤ 𝛿 (8)

with 𝛼1 and 𝛼2 can be chosen within the interval [0.5, 1]

and [1, 1.5] respectively. 𝛿1 and 𝛿2 are positive constant

numbers. From the above nonlinear structure, the feedback

gains are adjusted online depending on the value of the

error. For small position error, a large gain is produced, and

for large position error, a small gain is obtained. On the

other hand, large gains for large error rate and small gains

for small error rate. This behavior results with rapid

transition of the system and favorable damping. NPD is a

robust controller against the nonlinearities of PKMs,

parametric uncertainties and time delays.

D. Augmented Proportional Derivative Controller

APD, known also as PD+, is one of the conventional

model-based controllers composed of two main parts,

feedback part and dynamic model part. The feedback part

is a simple PD controller that guarantees the stability and

the dynamical part represents the nonlinear dynamics of the

system that compensates its effects and enhances the

control performance. The control law form of APD looks

as follows [19]:

𝜞 = 𝑴(𝒒)�̈�𝒅 + 𝑪(𝒒, �̇�)�̇�𝒅 + 𝑮(𝒒)

+ 𝜞𝑭𝒍𝒐𝒂𝒅+ 𝑲𝒑𝒆 + 𝑲𝒅�̇� (9)

As shown in equation (9), the dynamical term is computed

from the desired and actual trajectories. However, such

kind of controllers relying mostly on the dynamics of the

robot needs to have an accurate model information, and it

cannot compensate the effect of time-varying parameters

and uncertainties of PKMs.

E. Adaptive Feedforward with PD Controller

The AFFPD controller is quietly similar to the APD in the

general form, meaning that it is divided into two parts, one

part a simple PD feedback to conserve the stability and the

other part is the adaptive feedforward dynamics of the

PKM to reduce the influence of variation in parameters and

uncertainties. Thanks to the property of linearizing the

dynamic model, the adaptive term is the multiplication of

the regression matrix with the estimated vector of

parameters [20]. The control law equation is as follows:

𝜞 = 𝒀(𝒒𝒅, 𝒒�̇�, �̈�𝒅)�̂� + 𝑲𝒑𝒆 + 𝑲𝒅�̇� (10)

All the parameters ( �̂� 𝜖 ℝ6 ) need to be estimated and

adapted depending on the error. The controller relies on the

desired trajectories instead of the measured ones which can

improve the efficiency. The estimating algorithm is in

function of the measured error as follows:

�̇̂� = 𝑲 𝒀(. )𝑻𝝉𝑭𝑩 (11)

where 𝑲 𝜖 ℝ6×6 is a positive definite matrix that need to be

chosen for a good estimation and tracking error stability.

𝝉𝑭𝑩 is the torque computed from the feedback part. After

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The 1st International Congress for the Advancement of Mechanism, Machine, Robotics and Mechatronics Sciences, Beirut, Lebanon, 17-19 October, 2017

linearizing the inverse dynamic equation (3) considering all

the PKM parameters need to be estimated, the following

vector of parameters is obtained: Φ =[𝑀𝑇𝑃 𝑀𝑈𝑇𝑃 𝑀𝑠 𝐼𝑎 𝑚𝑎𝑟𝑎 𝑚𝑙𝑜𝑎𝑑]𝑻 such that MTP is the

total mass of the traveling plate including the contribution

of the forearms, MUTP is the mass of the upper traveling

plate including the contribution of the forearms, Ms is mass

of upper traveling plate with the equivalent mass to rotate

the screw. According to [23], the used adaptive control

scheme achieve a global asymptotic stability respecting the

necessary and sufficient conditions for adaptive control

[24], in which the reference trajectory should be chosen

rich enough with frequencies to converge the parameters

estimation’s error to zero, with a suitable initial values of

the parameters.

IV. Numerical Simulations and Experiments

VELOCE has four direct-drive motors TMB0140-

100-3RBS ETEL, they can provide maximum torque of

127 Nm and reach up a speed of 550 rpm. All actuators are

supplied with non-contact incremental optical encoders of

5000 pulses per revolution. The global structure can hold

as maximum payload of 10 Kg, achieve a peak velocity of

10m/s and peak acceleration of 200m/s2.

A. Simulation results

Simulations were done in Matlab/Simulink environment

implementing the controllers in discrete-time schemes

similar to real robots control. A fixed-step solver was

chosen of sample-time equal to 0.1 ms. The chosen desired

trajectory is a sequence of point-to-point motions with a

duration of each motion T= 0.5s. A nominal scenario of

motion is used to compare the performances of the

controllers such that no payload is considered in the

simulations. The evaluation criteria proposed to monitor

the performances is the computation of the Root Mean

Square Error over the Translational (RMSET) and

Rotational (RMSER) degrees-of-freedom as follow:

𝑅𝑀𝑆𝐸𝑇 = (1

𝑁∑ (𝑒𝑥

2(𝑖) + 𝑒𝑦2(𝑖) + 𝑒𝑧

2(𝑖))𝑁𝑖=1 )

1

2 (23)

𝑅𝑀𝑆𝐸𝑅 = (1

𝑁∑ (𝑒𝛼

2(𝑖))𝑁𝑖=1 )

1

2 (24)

where N is the number of the time-samples, 𝑒𝑥 , 𝑒𝑦 , 𝑒𝑧

represent the tracking errors along the axes x, y and z, 𝑒𝛼

Fig. 3. Evolution of the Cartesian tracking error in numerical simulation

PD / APD PID NPD AFFPD

𝑘𝑝 = 4000

𝑘𝑑 = 6

𝑘𝑝 = 4000

𝑘𝑑 = 6

𝑘𝑖 = 500

𝑘𝑝 = 2800

𝛼1 = 0.5

𝛿1 = 0.0062

𝑘𝑑 = 10

𝛼2 = 1.5

𝛿2 = 2.4131

𝑘𝑝 = 8000

𝑘𝑑 = 100

𝐾= 𝑑𝑖𝑎𝑔([100 100 5∗ 104 0.5 0.5 1])

Table 1. Control design gains in numerical simulation

denotes the tracking error along the rotational angle. The

gains for each controller in these simulations are specified

by the trial and error technique and shown in table 1. The

comparison between the three non-model-based

controllers, in fig. 3, shows that a NPD performs better than

the linear controllers (PD, PID). Thanks to its adjustable

gains with the error state, as discussed before, that grants it

more robustness and rejection for nonlinearity. For clarity,

a zoom in from 4 to 6 sec is done in the plot of the Cartesian

error in fig. 3, and the control input signals for the three

controllers are depicted in fig. 4. It is obvious that the

control input signals are within the allowable range that can

be handled by the real actuators. Similarly, the comparison

of the moving platform’s tracking error for the two model-

based controllers is presented in fig. 5. Apparently, the

benefit of parameters’ adaptation in the closed-loop of a

controller (AFFPD) is very significant in improving the

precision and accuracy, unlike the non-adaptive model-

based controller (APD) which is limited in rejecting the

uncertainties and parameters variation. Both control signals

are still under saturation and proper with the real actuators

limits (see Fig. 6). A good parameters’ estimation

convergence of the AFFPD controller is shown in fig. 7

reducing more the moving platform’s tracking error,

knowing that we initialize the parameters with much closed

values to the optimal numbers. The quantifications of the

errors all over the trajectory are shown in table 2 with the

improvements of each controller. It is notable to say that

the simulated model-based controllers are more accurate

than the non-model-based, as the calculation of the

percentages shows high improvements.

B. Experimental results

Due to its interesting specifications, AFFPD controller is

applied in real-time experiments on VELOCE robot, and

compared to the PD controller. The control architecture of

the VELOCE robot is implemented using Simulink from

Fig. 4. Evolution of the control input signals in numerical simulation

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The 1st International Congress for the Advancement of Mechanism, Machine, Robotics and Mechatronics Sciences, Beirut, Lebanon, 17-19 October, 2017

PD PID Improvement

on PD NPD

Improvement on PID

APD Improvement

on NPD AFFPD

Improvement on APD

RMSET

[cm] 0.0094 0.0078 17.02 % 0.0011 85.9 % 2.1018*10-5 98.09 % 1.0862*10-5 48.32 %

RMSER

[deg] 0.1309 0.0983 24.90 % 0.0147 85.04 % 1.6907*10-4 98.85 % 5.309*10-5 68.6 %

Table 2. Performance evaluation of the controllers in numerical simulation

Fig. 5. Evolution of the Cartesian tracking error in numerical simulation

Fig. 6. Evolution of the control input signals in numerical simulation

Fig. 7. Parameters estimation in numerical simulation of AFFPD

Mathworks Inc. and compiled using XPC Target (an

industrial computer of frequency 10 KHz i.e. the sample

time is 0.1 ms) and the Real-Time toolboxes. Same

Fig. 8. Evolution of the Cartesian tracking error in real-time experiments

Fig. 9. Evolution of the control input signals in real-time experiments

Fig. 10. Parameters estimation in real-time experiments of AFFPD

evaluation criteria used in simulations is considered in the

experiments. Retuning the gains of control design is needed

for experiments, and the obtained gains for AFFPD are:

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The 1st International Congress for the Advancement of Mechanism, Machine, Robotics and Mechatronics Sciences, Beirut, Lebanon, 17-19 October, 2017

PD AFFPD Improvements

RMSET [cm] 0.0156 0.0092 41.03 %

RMSER [deg] 1.077 0.7596 29.47 %

Table 3. Performance evaluation of PD and AFFPD controllers in real-

time experiments

𝑘𝑝 = 4000, 𝑘𝑑 = 6, and 𝐾 = 𝑑𝑖𝑎𝑔([2.5 ×

10−3 0.125 0.1 10−5 10−5 10−3]) . The plot of the

tracking error in Cartesian space for both controllers is

represented in fig. 8, showing the better and improved

global performance of AFFPD controller with respect to

PD controller. The evaluations and improvements in the

tracking error are computed and shown in table 3 validating

our pretend that model-based controllers are more powerful

than non-model-based controllers in real-time experiments.

More precisely, control schemes that include adaptive

dynamics provide robustness against parameters variation

and uncertainties. The control input signals of both

controllers are under saturation and in the safe range (see

Fig. 9). Figure 10 visualize a good convergence for the

estimated parameters in the AFFPD controller, which

contributes in minimizing the tracking error as possible.

One can notice the degradation of such root mean squares

of Cartesian tracking error of the two controllers from

numerical simulations to real-time experiments, and that is

normal because of the inaccurate model of PKMs exist in

the literature, in which they simplify friction, actuators’

dynamics, transmission system,…etc.

V. Conclusion and Future Work

In this paper, a comparison between the performances

of model-based (Augmented PD, Adaptive Feedforward

with PD) and non-model-based (PD, PID, Nonlinear PD)

controllers was done by numerical simulation sketching

and interpreting. We show the importance of including the

dynamic model of the PKMs in the closed-loop control, and

its main role in enhancing the performance of the

controller, especially when adapting the dynamical

parameters of the PKMs. Real-time experiments of PD and

AFFPD controllers were conducted on a 4-DOF parallel

robot to verify the validation of simulation results in the

real applications of parallel robots.

As a future perspective, one can look for more

accurate models of PKM involve the full dynamics such as

articulations’ friction, actuators’ dynamics, motor drivers,

and transmission system. Corporate these models in

adaptive closed-loop algorithms to improve the

performance of parallel robots, in terms of precision,

motion speed and robustness.

Acknowledgement

This paper has been supported by the Erasmus+

mobility project, ARPE ARROW project and the Lebanese

University.

References [1] Merlet, J.-P., Parallel Robots, Springer, The NetherLands, 2006. [2] Bennehar, M., Some Contributions to Nonlinear Adaptive

Control of PKMs : From Design to Real-Time Experiments,

PhD Thesis, University of Montpellier, 2015.

[3] Natal, G.S., Control of parallel robots: Towards very high

accelerations, PhD Thesis, University of Montpellier, 2016.

[4] Stewart, D., A platform with six degrees of freedom, Arch. Proc.

Inst. Mech. Eng. 1847-1982. 180 (1965) 371–386.

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